The Hardy number and the Bergman number of a planar domain are equal

TL;DR

This paper proves that for any planar domain, the Hardy number and the Bergman number are equal, linking the Hardy and weighted Bergman spaces through the domain's Green function and universal covering map.

Contribution

It improves a previous theorem by showing the equality of Hardy and Bergman numbers for all planar domains, connecting these function spaces more closely.

Findings

Hardy and Bergman numbers of a domain are equal

The Green function and universal covering map are key tools

The result generalizes previous theorems on function spaces

Abstract

This article deals with functions with a prefixed range and their inclusion in Hardy and weighted Bergman spaces. This idea was originally introduced by Hansen for Hardy spaces, and it was recently taken into weighted Bergman spaces by Karafyllia and Karamanlis. In particular, we improve a theorem of Karafyllia showing that the Hardy and Bergman numbers of any given domain coincide, that is, the Hardy and weighted Bergman spaces to which a function with prefixed range belongs can be related. The main tools in the proofs are the Green function of the domain and its universal covering map.